Geometric obstructions to fully ellipticity for families of manifolds with corners

TL;DR

This paper investigates index theory for families of manifolds with corners, revealing geometric obstructions to ellipticity and introducing a new combinatorial condition in the codimension 2 case related to conormal homology.

Contribution

It extends index theory for manifolds with corners by identifying a new topological obstruction in the codimension 2 case involving conormal homology cycles.

Findings

Identifies K-theoretical obstructions for elliptic families.

Provides a new combinatorial condition for codimension 2 corners.

Connects topological properties of the base with geometric obstructions.

Abstract

In the present paper we study index theory for families of manifold with corners. In particular the K-theoretical obstruction for an elliptic operator to have a family (Fredholm) index. For codimension 1 corners (families with boundary) it gives an expected condition on indices associated to codimension 1 faces. The main theorem of this paper concern the codimension 2 case: in addition to the expected condition on indices associated to codimension 2 faces, there is an extra combinatoric condition implying the topology of the family base. This new condition is expressed in terms of conormal homology cycles with K-theoretical coefficients.